nLab chromatic localization

Idea

The Bousfield localization of spectra $L_{E(n)}$ at $n$th Morava E-theory is called chromatic localization. The tower of chromatic localizations as $n$ ranges is called the chromatic tower, leading to the chromatic filtration. This is the subject of chromatic homotopy theory.

Properties

Relation to formal groups

Chromatic localization on complex oriented cohomology theories is like the restriction to the closed substack

$$\mathcal{M}_{FG}^{\leq n+1} \hookrightarrow \mathcal{M}_{FG} \times Spec \mathbb{Z}_{(p)}$$

of the moduli stack of formal groups on those of height $\geq n+1$.

(e.g. Lurie, lect 22, above theorem 1)

In this way the localization tower at the Morava E-theories exhibits the chromatic filtration in chromatic homotopy theory.

Examples

chromatic homotopy theory

chromatic level	complex oriented cohomology theory	E-∞ ring/A-∞ ring	real oriented cohomology theory
0	ordinary cohomology	Eilenberg-MacLane spectrum $H \mathbb{Z}$	HZR-theory
	0th Morava K-theory	$K(0)$
1	complex K-theory	complex K-theory spectrum $KU$	KR-theory
	first Morava K-theory	$K(1)$
	first Morava E-theory	$E(1)$
2	elliptic cohomology	elliptic spectrum $Ell_E$
	second Morava K-theory	$K(2)$
	second Morava E-theory	$E(2)$
	algebraic K-theory of KU	$K(KU)$
3 …10	K3 cohomology	K3 spectrum
$n$	$n$th Morava K-theory	$K(n)$
	$n$th Morava E-theory	$E(n)$	BPR-theory
$n+1$	algebraic K-theory applied to chrom. level $n$	$K(E_n)$ (red-shift conjecture)
$\infty$	complex cobordism cohomology	MU	MR-theory

Related concepts

• chromatic homotopy theory

References

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010 (lecture notes)

• Mark Hovey, Hal Sadofsky, Invertible spectra in the $E(n)$-local stable homotopy category (pdf)